Let's simplify each of the radicals one by one.

1. $\sqrt[3]{\frac{27x^6 y^9}{125x^3 y^5}}$

• First, simplify the numerical part: $\frac{27}{125}$. Since both are perfect cubes ($27 = 3^3$ and $125 = 5^3$), we get: $\sqrt[3]{\frac{27}{125}} = \frac{3}{5}$
• For the $x$-terms, $\frac{x^6}{x^3} = x^{6-3} = x^3$, and then taking the cube root gives $x$.
• For the $y$-terms, $\frac{y^9}{y^5} = y^{9-5} = y^4$, and taking the cube root gives $y^{4/3}$.

Thus, the simplified expression is: $\frac{3x y^{4/3}}{5}$

2. $\sqrt{64m^3n^4}$

• Simplify the square root of 64: $\sqrt{64} = 8$.
• For the $m$-terms: $\sqrt{m^3} = m^{3/2} = m\sqrt{m}$.
• For the $n$-terms: $\sqrt{n^4} = n^2$.

Thus, the simplified expression is: $8n^2 m\sqrt{m}$

3. $\sqrt{20a^{10}b^{20}c^9}$

• Simplify the numerical part: $\sqrt{20} = 2\sqrt{5}$.
• For the $a$-terms: $\sqrt{a^{10}} = a^5$.